Deficiency in infinite-dimensional manifolds

mstedam, TEteNetherkmds Received 30 November 1970

Abstrac : This paper is lqely devoted to plrovingthe following result: I.?: 8 be a F’r&het space home~morphk to its own countable infinite product,M be a pa ncompact m&fQid modeled on E, and lr:t K c R-2be closed. Then there exists a homeomc IThismof M onto IblX E taking K into 1MX (0) iff for each non-null, homotopically triviallope.aset U inltp, u\R: is aon-null and homotopically trivial.A similarresult for E = g2 (separtikieMilbbrtspace) is known.

AMSSubj. Class.: Primary5755; Secondary 5’701 r- Property::

L --

Edefkiency

open cQn(e

ambient invertibleisotopy J

1. Introduction

In [ 273 it w;ls shown that if E is a metric topological vector space (M’FVS) which is homedmorphic (z) to its own countable infinite product E” , and ikf is any E-manifold (i.e. M is a paracompact m&fold mcdeled on E:). then IMz nil X E. Accordingly we define a subset K of M to have E&,ficiency (or to be E-deficient) provided that K is closed and there exists a homeomorphism h : M + M X E such that h(K)cMx (0). Such sets hale proved to be important to the point-set topology of infinite dimensional manifolds because of results of the following two

types.

( 1) NegligibW, 1 theorems. If 122is separable infinite-dimensional IKilbert space:, 1Mis an ! *-manifold, and K C 1wis a countable union of E*-deficient sets, then it was shown in [ 11 that Arlz M\K. More general results have been est&lished in I[ $1. ’ Report ZW 19 70-010, DepartmentQf P Ire Mathematics,MathematicalCentre, Amsterdr;Pfn. 2 Researchpartially supportedby NSF G rantGP14429,

(2) Homeomorphism extension theorems. If M is as in(l), * *” ,

K1 and K* and h : K, + K2 is a horneomorphism which identity on K,), then &can be extended to a [ 6 1. For K, and K* additionally assumed to hals been established for more general linear

Q*-deficient sets in M, i? homotopic to id KP (the manifold homeomorphism be ANR’s, a similar result spacesthanQ* [ll]. In applications it is not easy to recognize that some sets have E-deficiency, thus it becomes desirable to have a coordinate-free topological characterization of E-deficiency in E-manifolds M. Such a characterization was obtained in [ 31 for M = @ and in 171 it was generalized to M any Q*-manifold. It states (using a notion introduced by Anderson in [ 33 ) that KC M (where M is an Q*-manifold) is Q*-deficient iff K has Property 2 (or is a Z-set), where a set P;‘in a space X has Property 2 iff F is closed and for each non-null, homotopically trivial open set U in X, U\F is non-null and homotopii=ally trivial. Among other things this enables us tchrecognize collared, closed sub-manifolds of M (i.e. boundaries of M) as being Q*-deficient and any closed subset of M which is a countable union of Q*-deficient sets is itself Q*-deficient. The main result of this paper is the following, which generalizes this characterization of E-deficiency to manifolds modeled on non-separab FrCchet spaces, where by a Freshet space we mean a locally convex complete MTVS. are

Theorem 1. Let E z EW be a Frechet space, 14 be an E-manifold, &nd let KC M be closed. Then K has E-deficiency iff K has Property 2. We remark that there are no known examples of infinite-dimensional Fr6chet spaces E which do not satisfy the condition E r, Ew. Concerning techniques it should be remarked that the proof of the correspondresult for M = @ [3] used the topollogy of the Hilbert cube I” and the fact that Q*can be compactified by I”
CA. Chapman, Deficiency in infinitdimemionul

265

mmifoids

(The notion of “arbitra .be made precise in the next seep e remark that by using different techniques David Iown (unpublished) that single Z-sets are str ‘olds, where E s P is a locally convex (LC) MTVS. In ‘i’heorem 2 we establish a homeomorphism extension theorem which generahzes the extension theorem of [CrJ (which was proved for @-manifolds) In Theorem 2’ below we give a simplified version of Theorem 2. T e more general statement appears in Section 5.

Ilwcrrem 2’. I ,ex E = Ea be a LCMTVS and let M be an E-manifold. If K, a1.d K2 art: E-deficient subsets of M and h : K1 + K2 is a homeomorphism w’hich is homotopic to idKl , then h can be extznded to a manlf’old ho:oleomorphism. We remark that II. Torgn&zyk has concurrently and independently obtained most of the results of this paper [ 181.

In this paper all spar/es will be assumed to be metric and all homeomorphisms GIll be assumed to be onto. Let _Yand Y be spaces aA let Q! be an open cover of Y. Then functions fpg : X -* Y are said to be Wcitw provided that for eavzhx E X there exists 2 U E Qklsuch that f(x), g(x) E U. A function I’ : X X I+ Y (where I = [ 0, I]) is said to be limited by ?l provided that for each x E X there exists a UE ?c such that F( (x} X i’) C U. If X is a sf:raceand F C X is closed, then by Lemma 3 of [S] there exists dn open cover ?c of X\F such that if h : X\F+ X\F is any homeomo,phi:j: n which is C&close to idXiF, ihen h can be extended to a homeomrJrphism h : X + X which satisfies ,%I F = idF. Such incover of X\F will be zalled normal (with respect to F). Let X be C.space and let {$} Tzl be a collection of homeomorphisms of X onto iulelf. Then for each x E X we let f(x) = lirni.+mfio -. 0./“&c), if this limit coxists.If f(x) exists, for a 1~ E X, tlwn we -wfite f = Ln~&y and call it the infinite left product of {fi}Tzl. We now state a convergence criterion for infinite 1p:ftproducts. This is essentially a reformulaa tion of West’s version [ 193 of Theorem 4.2 OF[4 I l

l

me mti

restx’itof this section is Thearera 3. Is where: we show that in certain spaces E-defici ncy and ~2-de~c~e~cy are equivalent concepts. A similar pr~p0siti~~ was established in [93, where E was additi~~~l~~ assumed to be a Banach space. We remark that the proof we give uf Theorem 3.I below fukluws in broad outline the proaf of the corresplodding result of [ 9 1, with ~ppr~p~at~ rn~d~~~atiu~s being made ta overcome the fack of a norm, We:will first need a technical result conce.rIzingopen ~~~~s. (By the ~~~~ Cofle over a space X (deputed by C(X)) we rnea~ the space (v u (X x (0, 1)), w~ch is t~pu~~giz~dby ~~~~~si~gas a basis the usu~1t~p~lugy un X x (0, f ) tugeth~r with aff sets of the f0n-n (u) ti (ATX (0, t)), fur al t E (S, I). We call tl the ~~~~e~ af the come.) We omit thie prouf of -&elemma, since it is similar to

~h~~r~rn 5.3 of

[ f7]

t

TVS, M be an E-man Ias E -c.eficiency if!‘ K has !P-deficiency. as Edefkiency isnd let f : M -+ 84 X E X E be a ~QRWI-

L&Es

eorem

[

E*

be a

151 we have E -z (-1,1)

EsE*Ez

.x G,

for some G. Thus

t-1, 1‘P X P’ z (-1,1l)* x (-i,l)*

X G” s (-1, .l)” x E.

since (-1,1)* = sz @ we IraveEr EX @. Letg:E+EX @ beahomeomorphism which satisks g(0) = (0,O) and let 7: M + IQ’X 14:X E X @* ~ede~nedby~(~)=(idX,~~of(*),whereidXg:MX~X~~MX~X L;’X Q2is defined by (id X g)(x, y, z) = (x, y, g(z)). Then 7’1: z &omeomorphism and f”(K) (1 A#X E X E X {0) . This. implies that q has @-deliciency. On the other hand. assume that FChas @deficiency. Thus we have a homeomorphism f : IM+ M X t2 such that f(K) C M X {O) * It iis easy to modify f to get a ho~nf~omorphism-7: M + M X E X [O, 11 X E which satisfies f(K)CM X E X (0) X E. (Use the fact that @ = .@ X IO, 1) [ 13 1.) Let h : 1T X [ (1, I.) X A!!? + C(E) X E be the horneomo~h~s~ desc~bedinLemma3.l.Then~d~X h:rlMIXEX [0,1)X E+MX l?(E)XE is a homeomo hism and (id,, X !I) Qf(K) C M X {v}X E,where v is the vertex of C:E). fnthe p~“oofof Lenlma 2 of [ B2 ] there is a proof that E X Q2s C(EX S1), where S, = ix E 92 1 IfxIt = 1) , Since S, z @ [ 131 we have E 2 C(E). Thus we c atI modify (idM X h) 0 f”to get a h.omeon~o~hism g : M + IMX E which s;atisfies g(K) .c M X {OiL

4, Deforming 8 manifold to an &deficient subset The n% result o FI-his section is Theorem 4.2, which shows how to deform cerr +I manifolds onto subsets which have II*-defkiency. The following Itk, Ia is neisded for its proof. Lem-ma4.1. Let .X be a space and let K be a closed subset of llrfX 1” such that K C X :< s, Then there exists a ho~neomor~~~isn~ f:XXI”jXX~“Hrhichsatisfiesf(K)cXX TI~zEl[-&~]q Proof. Foreach integer i > 0 and each x E A”let

glb (tit(~,t)E .x ) = &(

( 1, if K n

K},ifKn({x}

XI”)#

(pjf-3I”) =

where we adopt th,: convention that if t E , then ti is the ith coordi+ (-1, f 1 is lower seminate of t. It follows routinely that eachfi continuous. Thus by Dowker’s theorem ([ 101 p. 170) there is a contin+ (-1,1) which satisfies -1 < gi (-u)< f;:(x), for all x E X, Sinailady there is a continuous function g: : X + (-1,l) whiclh satisfies -1 C.g#) <: g#x) < 1 and ri 0 p2 (K n ((x} X I-)) C (gi (x), g:(x)), for all x E X, where ri : I” + Ii and p2 : X X I” + I” are projections. For each pair a, b, of real numbers satisfying -t < a < b < 1, there exists a unique piecewise linear homeomovhism h,, : [al, I] + [-I, 1 ] which satisfies !&&I) = -a 9 heea (b) = i, md h, #b is Pinear IR each of the intervals [&4], [a,& and [b, 11. Then define f:X x I"-+ XX 1"bY f(x,(ti)) = (x, ('gicxj gel), for dl (2, (ti)) E X X 1”. Clearly f fulfills OUf requirement;. ’ Theorem 4.1. Let E EEEW be a MTVS, A4*bean E-manifold, K C M be Q2deficient, and let W be an open covek of M. Then there exists a homotopy H = &I tEI : A4X ,I + A6such that Ho = id, Ht I K = id, for all t E I, HI : II4+ M is an embeidding such that HI(M) is Q2deficient, and Proof. Sinzs I” X s 2 s a-e can use an argument like that used in Lemma 6 of [T] to prove that a closed set F C IId has Q*defIciency iff exists a homeomorphism of AI onto M X I” taking F into M X (0). Thus let f : A#+ M X I” be rl homeomcrphism such that f(K) C 44 X (0). Using techniques like those used in Lemma 4.1 we can clearly construct a homotopy G : A4 X I” X i+ M X I” such that C, = id, G, : M X I”+ M X r is a closed embedding such that G @ X I-) C A! X s, 6, IA4X (0) = id, for all tp and G is limited by f(Q), the cover of A4X I” induced by f’and W. Then define H : M X I + M by If@) = f-l0 G,of(x). All w’ehave tc3do is show that H,(M) has Q2-deficiency. Note that fa N1(_A!)is a closed subset ofA X I” which satisfies X s. Ilsing Lemma 4.1 there exists a homsomorphism

comments above this proves that

e isolate a pat of the proof of in the proof of Tl-.ec,rem 1.

eorem 4.1 which will be useful

.l. LetEkEW bea TVS, M be an E-manifold, and let KcMXYbeac osed set such that K c M X s. Then K has !&deficiency.

5. A proof of Theorem 2 The main result of this section is Theorem 2, where we generalize the homeomorphism ebxtension theo m of [6] . Thz proof we give; follows in br Dad outline the proof given, m IS] , but the re are a tew technicalL ties which have to be overcome in order to make the proof work. We will need the following mapping replacement theorem which resembles Theorem 3.1 of [6].. Lemma 5.1. Let E z EW be a IXMTVS, M be an E-manifold, X be a

space which car be embedded as a closed subset of E, A c X be closed, and let f : X + M be a continuous function such thar f ; A is a homeomorphism of A omto an Edeficient subset of M. If 3d is any open cover of M, then there exists an embedding g :X + M such that g(X) is E-deficien:, gl A = flA, and g is ?C-close to f: Proof. Using Theorem 4.1 and the fact that E is an AR [ 161) a proof can be given which is similar to Theorem 3.1 of [ 6 ] . We will also need the following generalization of Theorem 2 of [a]. a 5.2. Let E 2 EW be a MTVS, M be a connected E-manifold, K C M be an E-deficient set. Then M canbeembedded as an open subset of E sothat K is taken onto an E-deficient (and therefore closed) subset of E. Proof. The proof p oceeds routinely as in Theorem 2 of [ ‘7 3 provided we note that (': )M can be embedded as an open subset ,DfE, and (2) there exists a homotopy H :E X I-, E such that H0 = idg , Ht is a homeomorphism (onto), for 0 < t < 1, and H1 :E + E\(O)isa homeomorphism. The firsit assertion is just Theorem 4 of f 121 and the second assertion follows since ;Pcorresponding property is tme for g2 [4] and also since E ssE x 112 (as was noted in Theorem 3.1).

EA. t%apmn, Deficiencyin iPtfmite-dimensi&wl martifokts

270

meorem 2. Let E SEiE;yjbe a LCMTVS, M

beanE-manifold, and let

be E-deficient subsets of M for which there exists a homotopy H ,: K,x I+ _!!4 such that Ho = idKl and H, :K '+K2 is a homeomorphism, If q is an open cover of M,such that k is limited by Cu, then there exists an an&ient invertible isotopy G :M X 1 + which satisfies C, = idM, 6,I K, = HI,amiG is limited by S@(Q), where StO(V) = 9 andSt~+l(?l)={u(UE~ iUn V#Q))lV~Stn(‘?L)~,rt>0.(An isotopy G :X X I + X is said to be an ambient iwertibk isotopy provided that each level is an onto homeomorphism and G * :X X I+ X, defi&ed by G?(x) = C;‘(X), is continuous.) Proof. First note that a homeomor,lhism extension heorem for E (without the limitationby covers) is easy to establish for E-deficient subsets of E.One merely uses the tech.nique of Klee [ 141, as used in [ 21 e Thus in the case that K, fj K3 = 0 we can use, Lemmas 5.1, S .2 and the %chniques of [6] to obtain our desired ambient invertible isotopy. Qn the, other hand assume that Kl n Kz # 0. The technique of Theorem 9 1 of [3] shows that K1 and K, are Z-sets, and therefore K, CJK, is a Z-set (arguing directly from tbe definition). Using an unpublished recjklltof David W. Henderson there exists, Sor each open cover %’ of M, a homotopy F: M X I+ M such that F0 =yid&l(F,(M)) n (K, u K2) := Q)(where Cl denotes closure), an(IIF is limited by %‘. Thus by Lemma 5.1 and an appropriate choice of Cu’, thert: exists an embedding F* : K2 X I+ M-such that F’$ = idKz’ F*(K, X I) is an E-deficient set ?I M for which Ff(K,) n (KI L) K2) = 0, and F* is limited by Cit. Using the above remarks there exists an zimbient invertible isotopy G* : M X I-, M such that G$ = idAr, GT 1K2 := F& and C* is limited by at. Note that K, and F,*(&) are disjoint E-deficient subsets of M and Ff 0 HI :ICI + F,U(&)is a homeomorphism which is homotoyjc to idK1, with a homotopy that is limited b:l St(9( ). We can once more use the above ,techniques to find an ambient invertible isofopy H * :M X I -+ JM such that H$ = iclM,HT 1K,= !Gfo H,,and H* is limiter! by S@(U). Then t31leobvious composition G =a(G*)- 1H Ok I’ulfills our reqr!iremen ts. i,,

K2

6. Proof of Theorem 1 The :jtep from E-de sembles Theorem 9.1 of [ 31. Property Z and let h : X presentation for I” an

I’”

perty 2 is straightforward and ree other implicaticrt! let Kc M have be a homeomorphism. Using the retion 2 let B(1") =

T.A. Chapman, Deficiency

in inj%aite-dimensiona~

manifokts

271

in [3] that: there is a hcmeomorphism of I” onto itself which sends B(I”‘) into
2T2

T.A. CZhupmn, Defidemy fnh~ite-dimensmm

nrtmifilds

[ 111 H&tdeqon, D .W., Stab&&s cation of infinitedimensional man!.foMsby homotopy type (sutrmitte& [IZ] Henderson, &I#., Corrections and extensions of two papersabout infiitedimencional w&foids (svbmM@). &131Kiee,V.L., Convey bodies and periodic homeom I&lbert space, Trans. Am. Mat@,$3~. 74 f1933)1043. ]14] Klee, XL., Some topological properties of convex se ans. %II Math. Sot. 78 (1955) 30-M. . [IS] Michael, E., Conw structures, and continuous selection, Car&. 3. Math. ll(1959) 556536. {ld] Pahis, IL, Homotopy theory of infinitedimensiorwl manifolds, I’o@ogy 5 (1966) l-16. [17] Schori, MM,, Topological stability for infinitedimensional manifolds, Compositio Math., to appear. [ iUS) Torundzyk, PI., (G,K)-skeleton and absorbing sets in complete metric spaces, to app5a. [ ll9] West, J.E., The ambient homosomorphy of an incomplete subspace PCinfinitedimensional i Hilbert space (submitted).